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Shared Qs (u11)


  1. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((94, 97)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-94)^2+(y_2-97)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (90,94)
    2 (38,55)
    3 (99,109)
    4 (64,57)
    5 (114,76)


    Solution


  2. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((102, 105)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-102)^2+(y_2-105)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (126,112)
    2 (142,114)
    3 (22,45)
    4 (120,81)
    5 (105,109)


    Solution


  3. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((103, 98)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-103)^2+(y_2-98)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (43,123)
    2 (43,178)
    3 (113,122)
    4 (52,30)
    5 (58,74)


    Solution


  4. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((91, 101)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-91)^2+(y_2-101)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (61,173)
    2 (131,92)
    3 (154,161)
    4 (103,66)
    5 (109,181)


    Solution


  5. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((104, 107)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-104)^2+(y_2-107)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (92,116)
    2 (83,35)
    3 (71,163)
    4 (124,59)
    5 (8,135)


    Solution


  6. Question

    Using Desmos, determine which of the given points are within a distance of 11 units from point (3, 5).

    1. Graph the inequality \((x-3)^2+(y-5)^2~\le~11^2\) as the first item in Desmos.
    2. Enter each point as a Cartesian coordinate pair. Actually, you should be able to copy/paste all the points as a single item (using commas to separate the points):
      • (12,11), (11,14), (7,16), (-1,14), (-5,12), (-8,2), (-7,1), (-2,-5), (8,-5), (15,2)
    3. Click “Label” to label the points.
    4. Determine which points are in the shaded area. If a point is exactly on the boundary, count that point as within the distance. You might need to zoom in on points to be sure.

    1. (12,11)
    2. (11,14)
    3. (7,16)
    4. (-1,14)
    5. (-5,12)
    6. (-8,2)
    7. (-7,1)
    8. (-2,-5)
    9. (8,-5)
    10. (15,2)

    Solution


  7. Question

    Using Desmos, determine which of the given points are within a distance of 11 units from point (6, 4).

    1. Graph the inequality \((x-6)^2+(y-4)^2~\le~11^2\) as the first item in Desmos.
    2. Enter each point as a Cartesian coordinate pair. Actually, you should be able to copy/paste all the points as a single item (using commas to separate the points):
      • (17,9), (9,13), (4,15), (-1,11), (-3,10), (-3,0), (-3,-5), (5,-7), (12,-6), (18,2)
    3. Click “Label” to label the points.
    4. Determine which points are in the shaded area. If a point is exactly on the boundary, count that point as within the distance. You might need to zoom in on points to be sure.

    1. (17,9)
    2. (9,13)
    3. (4,15)
    4. (-1,11)
    5. (-3,10)
    6. (-3,0)
    7. (-3,-5)
    8. (5,-7)
    9. (12,-6)
    10. (18,2)

    Solution


  8. Question

    Using Desmos, determine which of the given points are within a distance of 8 units from point (-6, 1).

    1. Graph the inequality \((x+6)^2+(y-1)^2~\le~8^2\) as the first item in Desmos.
    2. Enter each point as a Cartesian coordinate pair. Actually, you should be able to copy/paste all the points as a single item (using commas to separate the points):
      • (3,3), (-2,7), (-6,8), (-9,7), (-12,4), (-14,-2), (-9,-6), (-3,-8), (2,-2), (2,-1)
    3. Click “Label” to label the points.
    4. Determine which points are in the shaded area. If a point is exactly on the boundary, count that point as within the distance. You might need to zoom in on points to be sure.

    1. (3,3)
    2. (-2,7)
    3. (-6,8)
    4. (-9,7)
    5. (-12,4)
    6. (-14,-2)
    7. (-9,-6)
    8. (-3,-8)
    9. (2,-2)
    10. (2,-1)

    Solution


  9. Question

    Using Desmos, determine which of the given points are within a distance of 15 units from point (5, -1).

    1. Graph the inequality \((x-5)^2+(y+1)^2~\le~15^2\) as the first item in Desmos.
    2. Enter each point as a Cartesian coordinate pair. Actually, you should be able to copy/paste all the points as a single item (using commas to separate the points):
      • (18,9), (16,9), (4,14), (-6,10), (-10,-2), (-4,-13), (3,-17), (9,-14), (20,-7), (20,-3)
    3. Click “Label” to label the points.
    4. Determine which points are in the shaded area. If a point is exactly on the boundary, count that point as within the distance. You might need to zoom in on points to be sure.

    1. (18,9)
    2. (16,9)
    3. (4,14)
    4. (-6,10)
    5. (-10,-2)
    6. (-4,-13)
    7. (3,-17)
    8. (9,-14)
    9. (20,-7)
    10. (20,-3)

    Solution


  10. Question

    Using Desmos, determine which of the given points are within a distance of 8 units from point (-2, -3).

    1. Graph the inequality \((x+2)^2+(y+3)^2~\le~8^2\) as the first item in Desmos.
    2. Enter each point as a Cartesian coordinate pair. Actually, you should be able to copy/paste all the points as a single item (using commas to separate the points):
      • (6,-3), (4,5), (1,6), (-6,4), (-10,2), (-11,-4), (-9,-5), (-6,-9), (2,-10), (5,-8)
    3. Click “Label” to label the points.
    4. Determine which points are in the shaded area. If a point is exactly on the boundary, count that point as within the distance. You might need to zoom in on points to be sure.

    1. (6,-3)
    2. (4,5)
    3. (1,6)
    4. (-6,4)
    5. (-10,2)
    6. (-11,-4)
    7. (-9,-5)
    8. (-6,-9)
    9. (2,-10)
    10. (5,-8)

    Solution


  11. Question

    Consider the points listed below:

    Please determine which points satisfy both criteria below:

    1. Closer than 15 units from point (81, 41), and
    2. Closer than 15 units from point (61, 35).

    1. (68,40)
    2. (79,33)
    3. (78,31)
    4. (67,37)
    5. (62,34)
    6. (66,42)
    7. (77,29)
    8. (69,47)
    9. (73,39)
    10. (74,38)

    Solution


  12. Question

    Consider the points listed below:

    Please determine which points satisfy both criteria below:

    1. Closer than 19 units from point (35, 33), and
    2. Closer than 10 units from point (49, 17).

    1. (50,26)
    2. (36,23)
    3. (38,20)
    4. (42,17)
    5. (43,28)
    6. (40,25)
    7. (52,13)
    8. (41,27)
    9. (54,31)
    10. (48,21)

    Solution


  13. Question

    Consider the points listed below:

    Please determine which points satisfy both criteria below:

    1. Closer than 7 units from point (24, 47), and
    2. Closer than 13 units from point (35, 52).

    1. (30,50)
    2. (35,42)
    3. (23,41)
    4. (22,49)
    5. (21,39)
    6. (34,57)
    7. (24,51)
    8. (19,44)
    9. (36,52)
    10. (25,48)

    Solution


  14. Question

    Consider the points listed below:

    Please determine which points satisfy both criteria below:

    1. Closer than 22 units from point (29, 69), and
    2. Closer than 23 units from point (63, 49).

    1. (48,60)
    2. (46,63)
    3. (39,67)
    4. (45,57)
    5. (49,55)
    6. (47,62)
    7. (37,53)
    8. (41,58)
    9. (52,68)
    10. (36,52)

    Solution


  15. Question

    Consider the points listed below:

    Please determine which points satisfy both criteria below:

    1. Closer than 25 units from point (73, 31), and
    2. Closer than 24 units from point (39, 54).

    1. (61,40)
    2. (55,50)
    3. (48,37)
    4. (57,45)
    5. (53,49)
    6. (63,44)
    7. (62,34)
    8. (60,46)
    9. (58,48)
    10. (52,52)

    Solution


  16. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle in an \(xy\)-plane.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((142,163)\) \(80\)
    2 \((105,39)\) \(61\)
    3 \((34,162)\) \(87\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  17. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle in an \(xy\)-plane.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((167,79)\) \(75\)
    2 \((27,151)\) \(85\)
    3 \((111,70)\) \(34\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  18. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle in an \(xy\)-plane.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((164,180)\) \(100\)
    2 \((40,52)\) \(80\)
    3 \((119,64)\) \(39\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  19. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle in an \(xy\)-plane.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((88,128)\) \(25\)
    2 \((77,184)\) \(82\)
    3 \((35,59)\) \(75\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  20. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle in an \(xy\)-plane.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((136,124)\) \(34\)
    2 \((114,114)\) \(10\)
    3 \((154,72)\) \(60\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  21. Question

    Consider the triangle with vertices at the following coordinates:

    plot of chunk unnamed-chunk-1

    Calculate the perimeter. Please be accurate to within 0.01 units.


    Solution


  22. Question

    Consider the triangle with vertices at the following coordinates:

    plot of chunk unnamed-chunk-1

    Calculate the perimeter. Please be accurate to within 0.01 units.


    Solution


  23. Question

    Consider the triangle with vertices at the following coordinates:

    plot of chunk unnamed-chunk-1

    Calculate the perimeter. Please be accurate to within 0.01 units.


    Solution


  24. Question

    Consider the triangle with vertices at the following coordinates:

    plot of chunk unnamed-chunk-1

    Calculate the perimeter. Please be accurate to within 0.01 units.


    Solution


  25. Question

    Consider the triangle with vertices at the following coordinates:

    plot of chunk unnamed-chunk-1

    Calculate the perimeter. Please be accurate to within 0.01 units.


    Solution


  26. Question

    An archery target is 200 millimeters wide. It is composed of concentric rings. The center ring is worth 10 points, the next ring is worth 9, and so on as shown in the diagram. Notice the radii of the rings are all multiples of 10 millimeters.

    By setting up a Cartesian coordinate system, with units of millimeters, and the origin at the bullseye, we can indicate any point on the target with a Cartesian-coordinate ordered pair.

    plot of chunk unnamed-chunk-1

    An archer shoots 36 arrows; the positions are indicated in the table below.

    x (mm) y (mm)
    13 -15
    -19 -63
    16 -9
    -13 85
    -19 -24
    26 44
    23 19
    -55 -20
    -51 -17
    -43 -58
    44 31
    -50 1
    63 10
    -17 -37
    -22 55
    49 36
    -1 -59
    -38 21
    -30 11
    -14 -34
    -35 -6
    64 -1
    2 -17
    17 20
    61 -35
    -70 -11
    -46 27
    11 -37
    27 21
    35 -55
    -18 33
    -69 -20
    -21 15
    -7 20
    50 -25
    -29 -25


    How many of these arrows landed in the ring worth 5 points?

    (Assume the arrows are infinitely skinny. None landed exactly on a boundary.)


    Solution


  27. Question

    An archery target is 200 millimeters wide. It is composed of concentric rings. The center ring is worth 10 points, the next ring is worth 9, and so on as shown in the diagram. Notice the radii of the rings are all multiples of 10 millimeters.

    By setting up a Cartesian coordinate system, with units of millimeters, and the origin at the bullseye, we can indicate any point on the target with a Cartesian-coordinate ordered pair.

    plot of chunk unnamed-chunk-1

    An archer shoots 36 arrows; the positions are indicated in the table below.

    x (mm) y (mm)
    28 5
    -27 26
    -54 -19
    -15 8
    -51 36
    29 1
    12 35
    14 -66
    13 59
    37 5
    12 42
    61 34
    -12 6
    20 -47
    -62 11
    4 3
    68 34
    -76 12
    9 -66
    65 -48
    25 -11
    24 -62
    19 49
    19 -10
    10 -44
    29 40
    22 5
    -33 -24
    -26 -27
    -40 17
    -13 -12
    39 33
    26 -57
    -55 -36
    12 81
    17 -17


    How many of these arrows landed in the ring worth 4 points?

    (Assume the arrows are infinitely skinny. None landed exactly on a boundary.)


    Solution


  28. Question

    An archery target is 200 millimeters wide. It is composed of concentric rings. The center ring is worth 10 points, the next ring is worth 9, and so on as shown in the diagram. Notice the radii of the rings are all multiples of 10 millimeters.

    By setting up a Cartesian coordinate system, with units of millimeters, and the origin at the bullseye, we can indicate any point on the target with a Cartesian-coordinate ordered pair.

    plot of chunk unnamed-chunk-1

    An archer shoots 36 arrows; the positions are indicated in the table below.

    x (mm) y (mm)
    6 -4
    -66 42
    10 35
    20 90
    49 61
    38 -26
    -71 27
    44 -13
    67 -7
    49 -19
    5 15
    66 39
    0 -12
    61 6
    -3 -20
    34 69
    38 -4
    -11 5
    -7 26
    -40 -27
    0 -54
    -42 -75
    -38 -35
    -45 2
    11 10
    56 -19
    -51 -3
    59 10
    -23 20
    -12 25
    21 68
    5 -52
    3 3
    20 -20
    17 -27
    11 7


    How many of these arrows landed in the ring worth 5 points?

    (Assume the arrows are infinitely skinny. None landed exactly on a boundary.)


    Solution


  29. Question

    An archery target is 200 millimeters wide. It is composed of concentric rings. The center ring is worth 10 points, the next ring is worth 9, and so on as shown in the diagram. Notice the radii of the rings are all multiples of 10 millimeters.

    By setting up a Cartesian coordinate system, with units of millimeters, and the origin at the bullseye, we can indicate any point on the target with a Cartesian-coordinate ordered pair.

    plot of chunk unnamed-chunk-1

    An archer shoots 36 arrows; the positions are indicated in the table below.

    x (mm) y (mm)
    46 31
    67 57
    63 -30
    -3 -26
    -41 -55
    32 -32
    -34 18
    22 -19
    -8 -5
    14 47
    -34 64
    -10 -43
    46 -41
    -7 -71
    24 -52
    24 40
    -23 19
    39 17
    -6 2
    0 -51
    45 -32
    98 -10
    -22 -25
    17 21
    -38 4
    3 48
    24 7
    -2 -30
    -63 -54
    16 -17
    -11 43
    21 -27
    49 -46
    -23 -21
    -60 -15
    -25 13


    How many of these arrows landed in the ring worth 6 points?

    (Assume the arrows are infinitely skinny. None landed exactly on a boundary.)


    Solution


  30. Question

    An archery target is 200 millimeters wide. It is composed of concentric rings. The center ring is worth 10 points, the next ring is worth 9, and so on as shown in the diagram. Notice the radii of the rings are all multiples of 10 millimeters.

    By setting up a Cartesian coordinate system, with units of millimeters, and the origin at the bullseye, we can indicate any point on the target with a Cartesian-coordinate ordered pair.

    plot of chunk unnamed-chunk-1

    An archer shoots 36 arrows; the positions are indicated in the table below.

    x (mm) y (mm)
    6 -14
    27 3
    -40 21
    -31 -2
    5 18
    -18 17
    -43 52
    -11 65
    -61 45
    6 15
    50 18
    -21 -28
    0 74
    -11 -3
    14 -1
    -55 50
    -27 -39
    -39 50
    -16 8
    -14 17
    3 21
    -17 -82
    31 -51
    9 94
    6 41
    -30 86
    98 19
    -29 -24
    -16 15
    -79 31
    -58 62
    -45 62
    31 -5
    40 13
    1 9
    27 -88


    How many of these arrows landed in the ring worth 3 points?

    (Assume the arrows are infinitely skinny. None landed exactly on a boundary.)


    Solution


  31. Question

    A right rectangular prism is 19 inches wide, 11 inches deep, and 5 inches tall.

    What is the distance between opposite corners?

    plot of chunk unnamed-chunk-1

    Please be accurate within \(\pm\) 0.01 of the exact answer.


    Solution


  32. Question

    A right rectangular prism is 14 inches wide, 10 inches deep, and 5 inches tall.

    What is the distance between opposite corners?

    plot of chunk unnamed-chunk-1

    Please be accurate within \(\pm\) 0.01 of the exact answer.


    Solution


  33. Question

    A right rectangular prism is 17 inches wide, 19 inches deep, and 8 inches tall.

    What is the distance between opposite corners?

    plot of chunk unnamed-chunk-1

    Please be accurate within \(\pm\) 0.01 of the exact answer.


    Solution


  34. Question

    A right rectangular prism is 15 inches wide, 14 inches deep, and 17 inches tall.

    What is the distance between opposite corners?

    plot of chunk unnamed-chunk-1

    Please be accurate within \(\pm\) 0.01 of the exact answer.


    Solution


  35. Question

    A right rectangular prism is 12 inches wide, 9 inches deep, and 7 inches tall.

    What is the distance between opposite corners?

    plot of chunk unnamed-chunk-1

    Please be accurate within \(\pm\) 0.01 of the exact answer.


    Solution


  36. Question

    When plotting all \(x\)-\(y\) pairs that satisfy the given equation (make the equation true), the result is a circle.

    \[x^2+6x+y^2+10y~=~-18\]

    The standard form of an equation of a circle shows the center \((h,k)\) and radius \(r\) as parameters.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    In order to convert the given circle into standard form, you should complete the square (twice, once for \(x\)-containing terms and once for \(y\)-containing terms). After you convert the equation to standard form, indicate the values of the parameters.



    Solution


  37. Question

    When plotting all \(x\)-\(y\) pairs that satisfy the given equation (make the equation true), the result is a circle.

    \[x^2-16x+y^2-8y~=~-71\]

    The standard form of an equation of a circle shows the center \((h,k)\) and radius \(r\) as parameters.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    In order to convert the given circle into standard form, you should complete the square (twice, once for \(x\)-containing terms and once for \(y\)-containing terms). After you convert the equation to standard form, indicate the values of the parameters.



    Solution


  38. Question

    When plotting all \(x\)-\(y\) pairs that satisfy the given equation (make the equation true), the result is a circle.

    \[x^2-12x+y^2-8y~=~-43\]

    The standard form of an equation of a circle shows the center \((h,k)\) and radius \(r\) as parameters.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    In order to convert the given circle into standard form, you should complete the square (twice, once for \(x\)-containing terms and once for \(y\)-containing terms). After you convert the equation to standard form, indicate the values of the parameters.



    Solution


  39. Question

    When plotting all \(x\)-\(y\) pairs that satisfy the given equation (make the equation true), the result is a circle.

    \[x^2+12x+y^2-14y~=~-81\]

    The standard form of an equation of a circle shows the center \((h,k)\) and radius \(r\) as parameters.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    In order to convert the given circle into standard form, you should complete the square (twice, once for \(x\)-containing terms and once for \(y\)-containing terms). After you convert the equation to standard form, indicate the values of the parameters.



    Solution


  40. Question

    When plotting all \(x\)-\(y\) pairs that satisfy the given equation (make the equation true), the result is a circle.

    \[x^2+4x+y^2-8y~=~16\]

    The standard form of an equation of a circle shows the center \((h,k)\) and radius \(r\) as parameters.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    In order to convert the given circle into standard form, you should complete the square (twice, once for \(x\)-containing terms and once for \(y\)-containing terms). After you convert the equation to standard form, indicate the values of the parameters.



    Solution


  41. Question

    Consider the two points:

    A third collinear point, \(C\), is 40% of the way from point \(A\) to point \(B\), as shown below.

    plot of chunk unnamed-chunk-1

    Find the coordinates of point \(C\):

    (,)



    Solution


  42. Question

    Consider the two points:

    A third collinear point, \(C\), is 80% of the way from point \(A\) to point \(B\), as shown below.

    plot of chunk unnamed-chunk-1

    Find the coordinates of point \(C\):

    (,)



    Solution


  43. Question

    Consider the two points:

    A third collinear point, \(C\), is 40% of the way from point \(A\) to point \(B\), as shown below.

    plot of chunk unnamed-chunk-1

    Find the coordinates of point \(C\):

    (,)



    Solution


  44. Question

    Consider the two points:

    A third collinear point, \(C\), is 40% of the way from point \(A\) to point \(B\), as shown below.

    plot of chunk unnamed-chunk-1

    Find the coordinates of point \(C\):

    (,)



    Solution


  45. Question

    Consider the two points:

    A third collinear point, \(C\), is 40% of the way from point \(A\) to point \(B\), as shown below.

    plot of chunk unnamed-chunk-1

    Find the coordinates of point \(C\):

    (,)



    Solution


  46. Question

    A taxi service charges a fixed base cost of $2.72 and then a mileage cost of $1.75 per mile. The following people wonder whether they can travel a specified distance with the amount of cash they are carrying.



    Solution


  47. Question

    A taxi service charges a fixed base cost of $6.00 and then a mileage cost of $2.39 per mile. The following people wonder whether they can travel a specified distance with the amount of cash they are carrying.



    Solution


  48. Question

    A taxi service charges a fixed base cost of $7.64 and then a mileage cost of $2.90 per mile. The following people wonder whether they can travel a specified distance with the amount of cash they are carrying.



    Solution


  49. Question

    A taxi service charges a fixed base cost of $3.09 and then a mileage cost of $0.92 per mile. The following people wonder whether they can travel a specified distance with the amount of cash they are carrying.



    Solution


  50. Question

    A taxi service charges a fixed base cost of $2.79 and then a mileage cost of $1.34 per mile. The following people wonder whether they can travel a specified distance with the amount of cash they are carrying.



    Solution


  51. Question

    A thief is stealing xots and yivs. Each xot has a mass of 15 kilograms and a volume of 8 liters. Each yiv has a mass of 5 kilograms and a volume of 12 liters. The thief can carry a maximum mass of 75 kilograms and a maximum volume of 96 liters. The profit from each xot is $6.54 and the profit from each yiv is $8.78.

    For your convenience, those numbers are organized in the table below.

    attribute xot yiv capacity
    mass (kg) 15 5 75
    volume (L) 8 12 96
    profit ($) 6.54 8.78 \(\infty\)

    What is the maximum profit the thief can produce? (In dollars.)


    Solution


  52. Question

    A thief is stealing xots and yivs. Each xot has a mass of 16 kilograms and a volume of 12 liters. Each yiv has a mass of 8 kilograms and a volume of 9 liters. The thief can carry a maximum mass of 128 kilograms and a maximum volume of 108 liters. The profit from each xot is $2.05 and the profit from each yiv is $6.66.

    For your convenience, those numbers are organized in the table below.

    attribute xot yiv capacity
    mass (kg) 16 8 128
    volume (L) 12 9 108
    profit ($) 2.05 6.66 \(\infty\)

    What is the maximum profit the thief can produce? (In dollars.)


    Solution


  53. Question

    A thief is stealing xots and yivs. Each xot has a mass of 8 kilograms and a volume of 9 liters. Each yiv has a mass of 20 kilograms and a volume of 15 liters. The thief can carry a maximum mass of 160 kilograms and a maximum volume of 135 liters. The profit from each xot is $4.23 and the profit from each yiv is $8.94.

    For your convenience, those numbers are organized in the table below.

    attribute xot yiv capacity
    mass (kg) 8 20 160
    volume (L) 9 15 135
    profit ($) 4.23 8.94 \(\infty\)

    What is the maximum profit the thief can produce? (In dollars.)


    Solution


  54. Question

    A thief is stealing xots and yivs. Each xot has a mass of 18 kilograms and a volume of 20 liters. Each yiv has a mass of 6 kilograms and a volume of 4 liters. The thief can carry a maximum mass of 108 kilograms and a maximum volume of 80 liters. The profit from each xot is $1.29 and the profit from each yiv is $1.8.

    For your convenience, those numbers are organized in the table below.

    attribute xot yiv capacity
    mass (kg) 18 6 108
    volume (L) 20 4 80
    profit ($) 1.29 1.80 \(\infty\)

    What is the maximum profit the thief can produce? (In dollars.)


    Solution


  55. Question

    A thief is stealing xots and yivs. Each xot has a mass of 15 kilograms and a volume of 6 liters. Each yiv has a mass of 3 kilograms and a volume of 12 liters. The thief can carry a maximum mass of 45 kilograms and a maximum volume of 72 liters. The profit from each xot is $3.84 and the profit from each yiv is $5.17.

    For your convenience, those numbers are organized in the table below.

    attribute xot yiv capacity
    mass (kg) 15 3 45
    volume (L) 6 12 72
    profit ($) 3.84 5.17 \(\infty\)

    What is the maximum profit the thief can produce? (In dollars.)


    Solution


  56. Question

    Find the nearest distance from point \((-11,16)\) to line \(y=2x-22\).

    For flavor, imagine on a map, a river can be approximated by the line \(y=2x-22\). You are currently at position \((-11,16)\), and you want to know how far you need to travel to get to the river using the shortest path.

    plot of chunk unnamed-chunk-1

    What is the distance from the point to the line? Your answer should be accurate to within \(\pm 0.01\).


    Solution


  57. Question

    Find the nearest distance from point \((-14,-15)\) to line \(y=-3x+13\).

    For flavor, imagine on a map, a river can be approximated by the line \(y=-3x+13\). You are currently at position \((-14,-15)\), and you want to know how far you need to travel to get to the river using the shortest path.

    plot of chunk unnamed-chunk-1

    What is the distance from the point to the line? Your answer should be accurate to within \(\pm 0.01\).


    Solution


  58. Question

    Find the nearest distance from point \((16,-4)\) to line \(y=2x-1\).

    For flavor, imagine on a map, a river can be approximated by the line \(y=2x-1\). You are currently at position \((16,-4)\), and you want to know how far you need to travel to get to the river using the shortest path.

    plot of chunk unnamed-chunk-1

    What is the distance from the point to the line? Your answer should be accurate to within \(\pm 0.01\).


    Solution


  59. Question

    Find the nearest distance from point \((-17,16)\) to line \(y=\frac{-2}{3}x+22\).

    For flavor, imagine on a map, a river can be approximated by the line \(y=\frac{-2}{3}x+22\). You are currently at position \((-17,16)\), and you want to know how far you need to travel to get to the river using the shortest path.

    plot of chunk unnamed-chunk-1

    What is the distance from the point to the line? Your answer should be accurate to within \(\pm 0.01\).


    Solution


  60. Question

    Find the nearest distance from point \((-9,-23)\) to line \(y=-2x+4\).

    For flavor, imagine on a map, a river can be approximated by the line \(y=-2x+4\). You are currently at position \((-9,-23)\), and you want to know how far you need to travel to get to the river using the shortest path.

    plot of chunk unnamed-chunk-1

    What is the distance from the point to the line? Your answer should be accurate to within \(\pm 0.01\).


    Solution


  61. Question

    An event has two types of tickets: each adult ticket costs $5.83 and each child ticket costs $2.11. The event sold 48 tickets for a revenue of $153.36.


    Solution


  62. Question

    An event has two types of tickets: each adult ticket costs $5.52 and each child ticket costs $2.14. The event sold 39 tickets for a revenue of $144.30.


    Solution


  63. Question

    An event has two types of tickets: each adult ticket costs $7.29 and each child ticket costs $3.10. The event sold 63 tickets for a revenue of $274.91.


    Solution


  64. Question

    An event has two types of tickets: each adult ticket costs $6.32 and each child ticket costs $3.89. The event sold 48 tickets for a revenue of $213.45.


    Solution


  65. Question

    An event has two types of tickets: each adult ticket costs $6.77 and each child ticket costs $2.91. The event sold 73 tickets for a revenue of $405.43.


    Solution


  66. Question

    The inner dimensions of a frame are 39 inches by 16 inches. An artist hopes to print an image with an aspect ratio of 3.53 such that a uniformly wide mat fits around the image.

    plot of chunk unnamed-chunk-2

    Find \(b\), the width of the mat in inches. The tolerance is \(\pm 0.01\) inches.


    Solution


  67. Question

    The inner dimensions of a frame are 31 inches by 20 inches. An artist hopes to print an image with an aspect ratio of 2.33 such that a uniformly wide mat fits around the image.

    plot of chunk unnamed-chunk-2

    Find \(b\), the width of the mat in inches. The tolerance is \(\pm 0.01\) inches.


    Solution


  68. Question

    The inner dimensions of a frame are 39 inches by 30 inches. An artist hopes to print an image with an aspect ratio of 1.7 such that a uniformly wide mat fits around the image.

    plot of chunk unnamed-chunk-2

    Find \(y\), the height of the image in inches. The tolerance is \(\pm 0.01\) inches.


    Solution


  69. Question

    The inner dimensions of a frame are 29 inches by 23 inches. An artist hopes to print an image with an aspect ratio of 1.71 such that a uniformly wide mat fits around the image.

    plot of chunk unnamed-chunk-2

    Find \(b\), the width of the mat in inches. The tolerance is \(\pm 0.01\) inches.


    Solution


  70. Question

    The inner dimensions of a frame are 29 inches by 14 inches. An artist hopes to print an image with an aspect ratio of 4.07 such that a uniformly wide mat fits around the image.

    plot of chunk unnamed-chunk-2

    Find \(y\), the height of the image in inches. The tolerance is \(\pm 0.01\) inches.


    Solution


  71. Question

    A lifeguard wants to save a struggling swimmer as soon as possible. The lifeguard can run along the beach at 4 m/s and swim at 1 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(210-x)^2+60^2}\) meters.

    plot of chunk unnamed-chunk-2

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{4} + \frac{\sqrt{(210-x)^2+60^2}}{1}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.


    Solution


  72. Question

    A lifeguard wants to save a struggling swimmer as soon as possible. The lifeguard can run along the beach at 2.2 m/s and swim at 1.3 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(200-x)^2+30^2}\) meters.

    plot of chunk unnamed-chunk-2

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{2.2} + \frac{\sqrt{(200-x)^2+30^2}}{1.3}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.


    Solution


  73. Question

    A lifeguard wants to save a struggling swimmer as soon as possible. The lifeguard can run along the beach at 3.6 m/s and swim at 0.8 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(260-x)^2+40^2}\) meters.

    plot of chunk unnamed-chunk-2

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{3.6} + \frac{\sqrt{(260-x)^2+40^2}}{0.8}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.


    Solution


  74. Question

    A lifeguard wants to save a struggling swimmer as soon as possible. The lifeguard can run along the beach at 2.5 m/s and swim at 1.4 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(210-x)^2+30^2}\) meters.

    plot of chunk unnamed-chunk-2

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{2.5} + \frac{\sqrt{(210-x)^2+30^2}}{1.4}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.


    Solution


  75. Question

    A lifeguard wants to save a struggling swimmer as soon as possible. The lifeguard can run along the beach at 3.3 m/s and swim at 1.8 m/s. The lifeguard will run \(x\) meters and swim \(\sqrt{(270-x)^2+30^2}\) meters.

    plot of chunk unnamed-chunk-2

    Thus, the time is a function of \(x\).

    \[t(x) ~=~ \frac{x}{3.3} + \frac{\sqrt{(270-x)^2+30^2}}{1.8}\]

    Find the value of \(x\) (in meters) that minimizes the amount of time. The tolerance is \(\pm 0.1\) meters.


    Solution


  76. Question

    A student has taken 5 exams and gotten an average of 70.6. What score does the student need on the next exam to bring the average to 75? (All exams are equally weighted in the average.)


    Solution


  77. Question

    A student has taken 4 exams and gotten an average of 62.5. What score does the student need on the next exam to bring the average to 70? (All exams are equally weighted in the average.)


    Solution


  78. Question

    A student has taken 5 exams and gotten an average of 64.6. What score does the student need on the next exam to bring the average to 70? (All exams are equally weighted in the average.)


    Solution


  79. Question

    A student has taken 3 exams and gotten an average of 70. What score does the student need on the next exam to bring the average to 75? (All exams are equally weighted in the average.)


    Solution


  80. Question

    A student has taken 5 exams and gotten an average of 61.2. What score does the student need on the next exam to bring the average to 65? (All exams are equally weighted in the average.)


    Solution


  81. Question

    A circle is the collection of points equally distant from a center. If the circle has a center \((h,k)\) and a radius \(r\), then we can represent that circle in the \(xy\)-plane using the following standard form of a circle:

    \[(x-h)^2+(y-k)^2 = r^2\]

    Notice that this equation looks like the Pythagorean Equation (from the Pythagorean Theorem): \(a^2+b^2=c^2\). It also looks like the distance formula, were \(r\) is the distance, \(x-h\) is the horizontal displacement, and \(y-k\) is the vertical displacement.

    Consider the circle graphed below. Some of the points on the circle are \((12,-9)\), \((10,-7)\), \((8,-9)\), and \((10,-11)\).

    plot of chunk unnamed-chunk-1

    Find the values of the parameters \(h\), \(k\), and \(r\).



    Solution


  82. Question

    A circle is the collection of points equally distant from a center. If the circle has a center \((h,k)\) and a radius \(r\), then we can represent that circle in the \(xy\)-plane using the following standard form of a circle:

    \[(x-h)^2+(y-k)^2 = r^2\]

    Notice that this equation looks like the Pythagorean Equation (from the Pythagorean Theorem): \(a^2+b^2=c^2\). It also looks like the distance formula, were \(r\) is the distance, \(x-h\) is the horizontal displacement, and \(y-k\) is the vertical displacement.

    Consider the circle graphed below. Some of the points on the circle are \((11,-2)\), \((4,5)\), \((-3,-2)\), and \((4,-9)\).

    plot of chunk unnamed-chunk-1

    Find the values of the parameters \(h\), \(k\), and \(r\).



    Solution


  83. Question

    A circle is the collection of points equally distant from a center. If the circle has a center \((h,k)\) and a radius \(r\), then we can represent that circle in the \(xy\)-plane using the following standard form of a circle:

    \[(x-h)^2+(y-k)^2 = r^2\]

    Notice that this equation looks like the Pythagorean Equation (from the Pythagorean Theorem): \(a^2+b^2=c^2\). It also looks like the distance formula, were \(r\) is the distance, \(x-h\) is the horizontal displacement, and \(y-k\) is the vertical displacement.

    Consider the circle graphed below. Some of the points on the circle are \((5,-2)\), \((-4,7)\), \((-13,-2)\), and \((-4,-11)\).

    plot of chunk unnamed-chunk-1

    Find the values of the parameters \(h\), \(k\), and \(r\).



    Solution


  84. Question

    A circle is the collection of points equally distant from a center. If the circle has a center \((h,k)\) and a radius \(r\), then we can represent that circle in the \(xy\)-plane using the following standard form of a circle:

    \[(x-h)^2+(y-k)^2 = r^2\]

    Notice that this equation looks like the Pythagorean Equation (from the Pythagorean Theorem): \(a^2+b^2=c^2\). It also looks like the distance formula, were \(r\) is the distance, \(x-h\) is the horizontal displacement, and \(y-k\) is the vertical displacement.

    Consider the circle graphed below. Some of the points on the circle are \((14,-7)\), \((8,-1)\), \((2,-7)\), and \((8,-13)\).

    plot of chunk unnamed-chunk-1

    Find the values of the parameters \(h\), \(k\), and \(r\).



    Solution


  85. Question

    A circle is the collection of points equally distant from a center. If the circle has a center \((h,k)\) and a radius \(r\), then we can represent that circle in the \(xy\)-plane using the following standard form of a circle:

    \[(x-h)^2+(y-k)^2 = r^2\]

    Notice that this equation looks like the Pythagorean Equation (from the Pythagorean Theorem): \(a^2+b^2=c^2\). It also looks like the distance formula, were \(r\) is the distance, \(x-h\) is the horizontal displacement, and \(y-k\) is the vertical displacement.

    Consider the circle graphed below. Some of the points on the circle are \((-4,3)\), \((-9,8)\), \((-14,3)\), and \((-9,-2)\).

    plot of chunk unnamed-chunk-1

    Find the values of the parameters \(h\), \(k\), and \(r\).



    Solution


  86. Question

    The following equation produces a circle.

    \[(x-6)^2+(y+9)^2=121\]

    That circle has a center point, \((h,k)\), and radius, \(r\).

    Find the values of the parameters \(h\), \(k\), and \(r\).



    Solution


  87. Question

    The following equation produces a circle.

    \[(x+6)^2+(y-8)^2=144\]

    That circle has a center point, \((h,k)\), and radius, \(r\).

    Find the values of the parameters \(h\), \(k\), and \(r\).



    Solution


  88. Question

    The following equation produces a circle.

    \[(x-6)^2+(y+5)^2=144\]

    That circle has a center point, \((h,k)\), and radius, \(r\).

    Find the values of the parameters \(h\), \(k\), and \(r\).



    Solution


  89. Question

    The following equation produces a circle.

    \[(x+11)^2+(y-5)^2=64\]

    That circle has a center point, \((h,k)\), and radius, \(r\).

    Find the values of the parameters \(h\), \(k\), and \(r\).



    Solution


  90. Question

    The following equation produces a circle.

    \[(x+9)^2+(y-4)^2=100\]

    That circle has a center point, \((h,k)\), and radius, \(r\).

    Find the values of the parameters \(h\), \(k\), and \(r\).



    Solution


  91. Question

    A circle is centered at the origin with a radius equal to 1. This circle is called the unit circle, where “unit” refers to 1.

    A vertical line passes through point \((0.54,0.31)\).

    That vertical line intersects the circle at two points: one in Quadrant I and another in Quadrant IV.

    Determine the \(y\)-coordinate of the intersection between the line and the circle in Quadrant I.

    Please be accurate within \(\pm 0.001\).


    Solution


  92. Question

    A circle is centered at the origin with a radius equal to 1. This circle is called the unit circle, where “unit” refers to 1.

    A horizontal line passes through point \((0.54,0.25)\).

    That horizontal line intersects the circle at two points: one in Quadrant I and another in Quadrant II.

    Determine the \(x\)-coordinate of the intersection between the line and the circle in Quadrant I.

    Please be accurate within \(\pm 0.001\).


    Solution


  93. Question

    A circle is centered at the origin with a radius equal to 1. This circle is called the unit circle, where “unit” refers to 1.

    A vertical line passes through point \((0.23,0.64)\).

    That vertical line intersects the circle at two points: one in Quadrant I and another in Quadrant IV.

    Determine the \(y\)-coordinate of the intersection between the line and the circle in Quadrant I.

    Please be accurate within \(\pm 0.001\).


    Solution


  94. Question

    A circle is centered at the origin with a radius equal to 1. This circle is called the unit circle, where “unit” refers to 1.

    A horizontal line passes through point \((0.67,0.28)\).

    That horizontal line intersects the circle at two points: one in Quadrant I and another in Quadrant II.

    Determine the \(x\)-coordinate of the intersection between the line and the circle in Quadrant I.

    Please be accurate within \(\pm 0.001\).


    Solution


  95. Question

    A circle is centered at the origin with a radius equal to 1. This circle is called the unit circle, where “unit” refers to 1.

    A horizontal line passes through point \((0.69,0.32)\).

    That horizontal line intersects the circle at two points: one in Quadrant I and another in Quadrant II.

    Determine the \(x\)-coordinate of the intersection between the line and the circle in Quadrant I.

    Please be accurate within \(\pm 0.001\).


    Solution


  96. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-4.8,0)\) and \((4.8,0)\) and a covertex at \((0, 5.5)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (5.78, 3.36):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  97. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-7.7,0)\) and \((7.7,0)\) and a covertex at \((0, 3.6)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (3.86, 3.21):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  98. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-7.2,0)\) and \((7.2,0)\) and a covertex at \((0, 6.5)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (7, 4.5):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  99. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-2.4,0)\) and \((2.4,0)\) and a covertex at \((0, 7)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (5.24, 4.94):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  100. Question

    An ellipse is a set of points such that the total distance from (1) one focus to (2) the edge to (3) the other focus is constant for any point on the edge. The two foci (\(F_1\) and \(F_2\)) are a distance \(2c\) apart, and the total distance from \(F_1\) to the edge to \(F_2\) is a distance \(2a\).

    Let’s consider the example below with foci at \((-4.5,0)\) and \((4.5,0)\) and a covertex at \((0, 2.8)\):

    plot of chunk unnamed-chunk-2

    Evaluate \(2a\) and \(2c\).

    Now, let’s pick another point on the ellipse, with (approximate) coordinates (2.47, 2.48):

    plot of chunk unnamed-chunk-3

    The distance \(s\) is from \(F_1\) to the point on the edge. The distance \(w\) is from \(F_2\) to the point on the edge. Evaluate \(s\), \(w\), and the total distance (\(s+w\)).

    Remember, the total distance when connecting

    1. The first focus \(F_1\)
    2. Any point on the edge
    3. The second focus \(F_2\)

    equals \(2a\). Based on this information, find the location of the right vertex at (\(x\), 0) in the diagram below.

    plot of chunk unnamed-chunk-4



    Solution


  101. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+4)^2}{4}+\frac{(y+8)^2}{9}=1 & B&:~~\frac{(x+4)^2}{4}+\frac{(y-8)^2}{9}=1 & C&:~~\frac{(x-4)^2}{4}+\frac{(y+8)^2}{9}=1 & D&:~~\frac{(x-4)^2}{4}+\frac{(y-8)^2}{9}=1 \\ E&:~~\frac{(x+8)^2}{4}+\frac{(y+4)^2}{9}=1 & F&:~~\frac{(x+8)^2}{4}+\frac{(y-4)^2}{9}=1 & G&:~~\frac{(x-8)^2}{4}+\frac{(y+4)^2}{9}=1 & H&:~~\frac{(x-8)^2}{4}+\frac{(y-4)^2}{9}=1 \\ I&:~~\frac{(x+4)^2}{9}+\frac{(y+8)^2}{4}=1 & J&:~~\frac{(x+4)^2}{9}+\frac{(y-8)^2}{4}=1 & K&:~~\frac{(x-4)^2}{9}+\frac{(y+8)^2}{4}=1 & L&:~~\frac{(x-4)^2}{9}+\frac{(y-8)^2}{4}=1 \\ M&:~~\frac{(x+8)^2}{9}+\frac{(y+4)^2}{4}=1 & N&:~~\frac{(x+8)^2}{9}+\frac{(y-4)^2}{4}=1 & O&:~~\frac{(x-8)^2}{9}+\frac{(y+4)^2}{4}=1 & P&:~~\frac{(x-8)^2}{9}+\frac{(y-4)^2}{4}=1 \\ \end{align}\]

    Equation



    Solution


  102. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+3)^2}{36}+\frac{(y+4)^2}{49}=1 & B&:~~\frac{(x+3)^2}{36}+\frac{(y-4)^2}{49}=1 & C&:~~\frac{(x-3)^2}{36}+\frac{(y+4)^2}{49}=1 & D&:~~\frac{(x-3)^2}{36}+\frac{(y-4)^2}{49}=1 \\ E&:~~\frac{(x+4)^2}{36}+\frac{(y+3)^2}{49}=1 & F&:~~\frac{(x+4)^2}{36}+\frac{(y-3)^2}{49}=1 & G&:~~\frac{(x-4)^2}{36}+\frac{(y+3)^2}{49}=1 & H&:~~\frac{(x-4)^2}{36}+\frac{(y-3)^2}{49}=1 \\ I&:~~\frac{(x+3)^2}{49}+\frac{(y+4)^2}{36}=1 & J&:~~\frac{(x+3)^2}{49}+\frac{(y-4)^2}{36}=1 & K&:~~\frac{(x-3)^2}{49}+\frac{(y+4)^2}{36}=1 & L&:~~\frac{(x-3)^2}{49}+\frac{(y-4)^2}{36}=1 \\ M&:~~\frac{(x+4)^2}{49}+\frac{(y+3)^2}{36}=1 & N&:~~\frac{(x+4)^2}{49}+\frac{(y-3)^2}{36}=1 & O&:~~\frac{(x-4)^2}{49}+\frac{(y+3)^2}{36}=1 & P&:~~\frac{(x-4)^2}{49}+\frac{(y-3)^2}{36}=1 \\ \end{align}\]

    Equation



    Solution


  103. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+2)^2}{25}+\frac{(y+3)^2}{36}=1 & B&:~~\frac{(x+2)^2}{25}+\frac{(y-3)^2}{36}=1 & C&:~~\frac{(x-2)^2}{25}+\frac{(y+3)^2}{36}=1 & D&:~~\frac{(x-2)^2}{25}+\frac{(y-3)^2}{36}=1 \\ E&:~~\frac{(x+3)^2}{25}+\frac{(y+2)^2}{36}=1 & F&:~~\frac{(x+3)^2}{25}+\frac{(y-2)^2}{36}=1 & G&:~~\frac{(x-3)^2}{25}+\frac{(y+2)^2}{36}=1 & H&:~~\frac{(x-3)^2}{25}+\frac{(y-2)^2}{36}=1 \\ I&:~~\frac{(x+2)^2}{36}+\frac{(y+3)^2}{25}=1 & J&:~~\frac{(x+2)^2}{36}+\frac{(y-3)^2}{25}=1 & K&:~~\frac{(x-2)^2}{36}+\frac{(y+3)^2}{25}=1 & L&:~~\frac{(x-2)^2}{36}+\frac{(y-3)^2}{25}=1 \\ M&:~~\frac{(x+3)^2}{36}+\frac{(y+2)^2}{25}=1 & N&:~~\frac{(x+3)^2}{36}+\frac{(y-2)^2}{25}=1 & O&:~~\frac{(x-3)^2}{36}+\frac{(y+2)^2}{25}=1 & P&:~~\frac{(x-3)^2}{36}+\frac{(y-2)^2}{25}=1 \\ \end{align}\]

    Equation



    Solution


  104. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+2)^2}{16}+\frac{(y+5)^2}{64}=1 & B&:~~\frac{(x+2)^2}{16}+\frac{(y-5)^2}{64}=1 & C&:~~\frac{(x-2)^2}{16}+\frac{(y+5)^2}{64}=1 & D&:~~\frac{(x-2)^2}{16}+\frac{(y-5)^2}{64}=1 \\ E&:~~\frac{(x+5)^2}{16}+\frac{(y+2)^2}{64}=1 & F&:~~\frac{(x+5)^2}{16}+\frac{(y-2)^2}{64}=1 & G&:~~\frac{(x-5)^2}{16}+\frac{(y+2)^2}{64}=1 & H&:~~\frac{(x-5)^2}{16}+\frac{(y-2)^2}{64}=1 \\ I&:~~\frac{(x+2)^2}{64}+\frac{(y+5)^2}{16}=1 & J&:~~\frac{(x+2)^2}{64}+\frac{(y-5)^2}{16}=1 & K&:~~\frac{(x-2)^2}{64}+\frac{(y+5)^2}{16}=1 & L&:~~\frac{(x-2)^2}{64}+\frac{(y-5)^2}{16}=1 \\ M&:~~\frac{(x+5)^2}{64}+\frac{(y+2)^2}{16}=1 & N&:~~\frac{(x+5)^2}{64}+\frac{(y-2)^2}{16}=1 & O&:~~\frac{(x-5)^2}{64}+\frac{(y+2)^2}{16}=1 & P&:~~\frac{(x-5)^2}{64}+\frac{(y-2)^2}{16}=1 \\ \end{align}\]

    Equation



    Solution


  105. Question

    An ellipse is the set of (\(x\), \(y\)) points that satisfy the equation below.

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    where the center is \((h,k)\), the horizontal radius is \(r_1\), and the vertical radius is \(r_2\).

    Determine the parameters of the ellipse in the graph below. You can assume all parameters are integers.

    plot of chunk unnamed-chunk-2

    Which of the following equations would give the graph above?

    \[\begin{align} A&:~~\frac{(x+3)^2}{4}+\frac{(y+4)^2}{49}=1 & B&:~~\frac{(x+3)^2}{4}+\frac{(y-4)^2}{49}=1 & C&:~~\frac{(x-3)^2}{4}+\frac{(y+4)^2}{49}=1 & D&:~~\frac{(x-3)^2}{4}+\frac{(y-4)^2}{49}=1 \\ E&:~~\frac{(x+4)^2}{4}+\frac{(y+3)^2}{49}=1 & F&:~~\frac{(x+4)^2}{4}+\frac{(y-3)^2}{49}=1 & G&:~~\frac{(x-4)^2}{4}+\frac{(y+3)^2}{49}=1 & H&:~~\frac{(x-4)^2}{4}+\frac{(y-3)^2}{49}=1 \\ I&:~~\frac{(x+3)^2}{49}+\frac{(y+4)^2}{4}=1 & J&:~~\frac{(x+3)^2}{49}+\frac{(y-4)^2}{4}=1 & K&:~~\frac{(x-3)^2}{49}+\frac{(y+4)^2}{4}=1 & L&:~~\frac{(x-3)^2}{49}+\frac{(y-4)^2}{4}=1 \\ M&:~~\frac{(x+4)^2}{49}+\frac{(y+3)^2}{4}=1 & N&:~~\frac{(x+4)^2}{49}+\frac{(y-3)^2}{4}=1 & O&:~~\frac{(x-4)^2}{49}+\frac{(y+3)^2}{4}=1 & P&:~~\frac{(x-4)^2}{49}+\frac{(y-3)^2}{4}=1 \\ \end{align}\]

    Equation



    Solution


  106. Question

    The following equation (in polynomial form) represents an ellipse.

    \[36x^2+216x+4y^2+32y+244=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  107. Question

    The following equation (in polynomial form) represents an ellipse.

    \[25x^2+150x+4y^2+8y+129=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  108. Question

    The following equation (in polynomial form) represents an ellipse.

    \[64x^2+768x+4y^2-8y+2052=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  109. Question

    The following equation (in polynomial form) represents an ellipse.

    \[9x^2+90x+4y^2+48y+333=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  110. Question

    The following equation (in polynomial form) represents an ellipse.

    \[16x^2+32x+49y^2-490y+457=0\]

    With some algebra (completing the square), you can convert the equation into standard form:

    \[\frac{(x-h)^2}{(r_1)^2}+\frac{(y-k)^2}{(r_2)^2}=1\]

    From the given polynomial form, determine the 4 parameters.



    Solution


  111. Question

    An alien creature flies a round trip — from home, to school, and back home — using the exact same route both directions (so the distances of the initial and return trips are equal).

    On the initial trip, the creature maintained a speed of 84 meters per second. On the return trip, the creature maintained a speed of 51 meters per second.

    What was the average speed, in m/s, of the entire trip? (The tolerance is \(\pm 0.01\) m/s.)


    Solution


  112. Question

    An alien creature flies a round trip — from home, to school, and back home — using the exact same route both directions (so the distances of the initial and return trips are equal).

    On the initial trip, the creature maintained a speed of 24 meters per second. On the return trip, the creature maintained a speed of 62 meters per second.

    What was the average speed, in m/s, of the entire trip? (The tolerance is \(\pm 0.01\) m/s.)


    Solution


  113. Question

    An alien creature flies a round trip — from home, to school, and back home — using the exact same route both directions (so the distances of the initial and return trips are equal).

    On the initial trip, the creature maintained a speed of 65 meters per second. On the return trip, the creature maintained a speed of 80 meters per second.

    What was the average speed, in m/s, of the entire trip? (The tolerance is \(\pm 0.01\) m/s.)


    Solution


  114. Question

    An alien creature flies a round trip — from home, to school, and back home — using the exact same route both directions (so the distances of the initial and return trips are equal).

    On the initial trip, the creature maintained a speed of 88 meters per second. On the return trip, the creature maintained a speed of 73 meters per second.

    What was the average speed, in m/s, of the entire trip? (The tolerance is \(\pm 0.01\) m/s.)


    Solution


  115. Question

    An alien creature flies a round trip — from home, to school, and back home — using the exact same route both directions (so the distances of the initial and return trips are equal).

    On the initial trip, the creature maintained a speed of 39 meters per second. On the return trip, the creature maintained a speed of 35 meters per second.

    What was the average speed, in m/s, of the entire trip? (The tolerance is \(\pm 0.01\) m/s.)


    Solution


  116. Question

    A Lagrange polynomial is a polynomial that exactly hits all points from a list.

    Let’s say we have three points: \((x_1,y_1)\) and \((x_2,y_2)\) and \((x_3,y_3)\). The Lagrange polynomial is:

    \[L(x)~~=~~y_1\cdot\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}~+~y_2\cdot\frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)}~+~y_3\cdot\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}\]

    You can probably guess correctly how this would generalize with more points, but this seems tedious enough. Eh, what the heck… if there were four points, it would look like:

    \[L(x)~~=~~y_1\cdot\frac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)}~+~y_2\cdot\frac{(x-x_1)(x-x_3)(x-x_4)}{(x_2-x_1)(x_2-x_3)(x_2-x_4)}~+~y_3\cdot\frac{(x-x_1)(x-x_2)(x-x_4)}{(x_3-x_1)(x_3-x_2)(x_3-x_4)}~+~y_4\cdot\frac{(x-x_1)(x-x_2)(x-x_3)}{(x_4-x_1)(x_4-x_2)(x_4-x_3)}\]

    LOL. Anyway, I want you to find the Lagrange polynomial from the following list of three points.

    \[(5,9)\] \[(6,2)\] \[(7,4)\]

    First define \(L(x)\). Then, evaluate \(L(3)\). The tolerance is \(\pm 0.01\).


    Solution


  117. Question

    A Lagrange polynomial is a polynomial that exactly hits all points from a list.

    Let’s say we have three points: \((x_1,y_1)\) and \((x_2,y_2)\) and \((x_3,y_3)\). The Lagrange polynomial is:

    \[L(x)~~=~~y_1\cdot\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}~+~y_2\cdot\frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)}~+~y_3\cdot\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}\]

    You can probably guess correctly how this would generalize with more points, but this seems tedious enough. Eh, what the heck… if there were four points, it would look like:

    \[L(x)~~=~~y_1\cdot\frac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)}~+~y_2\cdot\frac{(x-x_1)(x-x_3)(x-x_4)}{(x_2-x_1)(x_2-x_3)(x_2-x_4)}~+~y_3\cdot\frac{(x-x_1)(x-x_2)(x-x_4)}{(x_3-x_1)(x_3-x_2)(x_3-x_4)}~+~y_4\cdot\frac{(x-x_1)(x-x_2)(x-x_3)}{(x_4-x_1)(x_4-x_2)(x_4-x_3)}\]

    LOL. Anyway, I want you to find the Lagrange polynomial from the following list of three points.

    \[(2,3)\] \[(6,5)\] \[(8,9)\]

    First define \(L(x)\). Then, evaluate \(L(4)\). The tolerance is \(\pm 0.01\).


    Solution


  118. Question

    A Lagrange polynomial is a polynomial that exactly hits all points from a list.

    Let’s say we have three points: \((x_1,y_1)\) and \((x_2,y_2)\) and \((x_3,y_3)\). The Lagrange polynomial is:

    \[L(x)~~=~~y_1\cdot\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}~+~y_2\cdot\frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)}~+~y_3\cdot\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}\]

    You can probably guess correctly how this would generalize with more points, but this seems tedious enough. Eh, what the heck… if there were four points, it would look like:

    \[L(x)~~=~~y_1\cdot\frac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)}~+~y_2\cdot\frac{(x-x_1)(x-x_3)(x-x_4)}{(x_2-x_1)(x_2-x_3)(x_2-x_4)}~+~y_3\cdot\frac{(x-x_1)(x-x_2)(x-x_4)}{(x_3-x_1)(x_3-x_2)(x_3-x_4)}~+~y_4\cdot\frac{(x-x_1)(x-x_2)(x-x_3)}{(x_4-x_1)(x_4-x_2)(x_4-x_3)}\]

    LOL. Anyway, I want you to find the Lagrange polynomial from the following list of three points.

    \[(4,2)\] \[(5,6)\] \[(8,3)\]

    First define \(L(x)\). Then, evaluate \(L(9)\). The tolerance is \(\pm 0.01\).


    Solution


  119. Question

    A Lagrange polynomial is a polynomial that exactly hits all points from a list.

    Let’s say we have three points: \((x_1,y_1)\) and \((x_2,y_2)\) and \((x_3,y_3)\). The Lagrange polynomial is:

    \[L(x)~~=~~y_1\cdot\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}~+~y_2\cdot\frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)}~+~y_3\cdot\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}\]

    You can probably guess correctly how this would generalize with more points, but this seems tedious enough. Eh, what the heck… if there were four points, it would look like:

    \[L(x)~~=~~y_1\cdot\frac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)}~+~y_2\cdot\frac{(x-x_1)(x-x_3)(x-x_4)}{(x_2-x_1)(x_2-x_3)(x_2-x_4)}~+~y_3\cdot\frac{(x-x_1)(x-x_2)(x-x_4)}{(x_3-x_1)(x_3-x_2)(x_3-x_4)}~+~y_4\cdot\frac{(x-x_1)(x-x_2)(x-x_3)}{(x_4-x_1)(x_4-x_2)(x_4-x_3)}\]

    LOL. Anyway, I want you to find the Lagrange polynomial from the following list of three points.

    \[(2,8)\] \[(5,9)\] \[(7,6)\]

    First define \(L(x)\). Then, evaluate \(L(4)\). The tolerance is \(\pm 0.01\).


    Solution


  120. Question

    A Lagrange polynomial is a polynomial that exactly hits all points from a list.

    Let’s say we have three points: \((x_1,y_1)\) and \((x_2,y_2)\) and \((x_3,y_3)\). The Lagrange polynomial is:

    \[L(x)~~=~~y_1\cdot\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}~+~y_2\cdot\frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)}~+~y_3\cdot\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}\]

    You can probably guess correctly how this would generalize with more points, but this seems tedious enough. Eh, what the heck… if there were four points, it would look like:

    \[L(x)~~=~~y_1\cdot\frac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)}~+~y_2\cdot\frac{(x-x_1)(x-x_3)(x-x_4)}{(x_2-x_1)(x_2-x_3)(x_2-x_4)}~+~y_3\cdot\frac{(x-x_1)(x-x_2)(x-x_4)}{(x_3-x_1)(x_3-x_2)(x_3-x_4)}~+~y_4\cdot\frac{(x-x_1)(x-x_2)(x-x_3)}{(x_4-x_1)(x_4-x_2)(x_4-x_3)}\]

    LOL. Anyway, I want you to find the Lagrange polynomial from the following list of three points.

    \[(3,2)\] \[(5,7)\] \[(6,4)\]

    First define \(L(x)\). Then, evaluate \(L(9)\). The tolerance is \(\pm 0.01\).


    Solution


  121. Question

    In Desmos, we can get best-fit curves by using their least-squares regression tool. This type of analysis is very important in science and research.

    In this problem, I want you to run a cubic regression on the following points:

    x y
    3 7
    5 6
    6 7
    7 8
    8 4

    The result of the regression will be a cubic polynomial, which we can call \(f\). You will then evaluate \(f(9)\).

    If you copy/paste the above table into a Desmos item, it should create a table. If not, add a new table and add the points shown above.

    In that item’s top-left corner, there should be a “Add Regression” button; click it. Change the regression type to “Cubic Regression”. You’ll see a cubic curve that does its best to get as close to the points as possible. You can “Export a snapshot to the expression list” by clicking on the button next to EQUATION. In that new expression, replace the \(y\) with \(f(x)\).

    Now, in a new item, you should be able to evaluate \(f(9)\). What do you get? (We are using regression to make a prediction. The tolerance is \(\pm0.01\).)


    Solution


  122. Question

    In Desmos, we can get best-fit curves by using their least-squares regression tool. This type of analysis is very important in science and research.

    In this problem, I want you to run a cubic regression on the following points:

    x y
    3 1
    5 2
    7 3
    8 3
    9 6

    The result of the regression will be a cubic polynomial, which we can call \(f\). You will then evaluate \(f(1)\).

    If you copy/paste the above table into a Desmos item, it should create a table. If not, add a new table and add the points shown above.

    In that item’s top-left corner, there should be a “Add Regression” button; click it. Change the regression type to “Cubic Regression”. You’ll see a cubic curve that does its best to get as close to the points as possible. You can “Export a snapshot to the expression list” by clicking on the button next to EQUATION. In that new expression, replace the \(y\) with \(f(x)\).

    Now, in a new item, you should be able to evaluate \(f(1)\). What do you get? (We are using regression to make a prediction. The tolerance is \(\pm0.01\).)


    Solution


  123. Question

    In Desmos, we can get best-fit curves by using their least-squares regression tool. This type of analysis is very important in science and research.

    In this problem, I want you to run a cubic regression on the following points:

    x y
    1 6
    2 1
    3 5
    4 8
    7 9

    The result of the regression will be a cubic polynomial, which we can call \(f\). You will then evaluate \(f(6)\).

    If you copy/paste the above table into a Desmos item, it should create a table. If not, add a new table and add the points shown above.

    In that item’s top-left corner, there should be a “Add Regression” button; click it. Change the regression type to “Cubic Regression”. You’ll see a cubic curve that does its best to get as close to the points as possible. You can “Export a snapshot to the expression list” by clicking on the button next to EQUATION. In that new expression, replace the \(y\) with \(f(x)\).

    Now, in a new item, you should be able to evaluate \(f(6)\). What do you get? (We are using regression to make a prediction. The tolerance is \(\pm0.01\).)


    Solution


  124. Question

    In Desmos, we can get best-fit curves by using their least-squares regression tool. This type of analysis is very important in science and research.

    In this problem, I want you to run a cubic regression on the following points:

    x y
    2 8
    3 1
    5 2
    6 2
    8 5

    The result of the regression will be a cubic polynomial, which we can call \(f\). You will then evaluate \(f(1)\).

    If you copy/paste the above table into a Desmos item, it should create a table. If not, add a new table and add the points shown above.

    In that item’s top-left corner, there should be a “Add Regression” button; click it. Change the regression type to “Cubic Regression”. You’ll see a cubic curve that does its best to get as close to the points as possible. You can “Export a snapshot to the expression list” by clicking on the button next to EQUATION. In that new expression, replace the \(y\) with \(f(x)\).

    Now, in a new item, you should be able to evaluate \(f(1)\). What do you get? (We are using regression to make a prediction. The tolerance is \(\pm0.01\).)


    Solution


  125. Question

    In Desmos, we can get best-fit curves by using their least-squares regression tool. This type of analysis is very important in science and research.

    In this problem, I want you to run a cubic regression on the following points:

    x y
    2 3
    3 4
    5 5
    6 4
    8 3

    The result of the regression will be a cubic polynomial, which we can call \(f\). You will then evaluate \(f(1)\).

    If you copy/paste the above table into a Desmos item, it should create a table. If not, add a new table and add the points shown above.

    In that item’s top-left corner, there should be a “Add Regression” button; click it. Change the regression type to “Cubic Regression”. You’ll see a cubic curve that does its best to get as close to the points as possible. You can “Export a snapshot to the expression list” by clicking on the button next to EQUATION. In that new expression, replace the \(y\) with \(f(x)\).

    Now, in a new item, you should be able to evaluate \(f(1)\). What do you get? (We are using regression to make a prediction. The tolerance is \(\pm0.01\).)


    Solution


  126. Question

    x y
    39.2 283.5
    39.1 291.8
    31.8 234.5
    38.0 268.1
    38.4 286.8
    36.2 253.7
    36.4 249.1

    Find the trendline using a simple linear regression. Do not assume the \(y\)-intercept is 0.

    Using the linear model, predict \(y\) when \(x=41\).


    Solution


  127. Question

    x y
    65.1 282.3
    61.8 279.0
    63.2 281.5
    60.0 268.1
    57.7 255.9
    62.6 272.0
    63.9 274.7
    59.1 262.3
    66.5 292.9

    Find the trendline using a simple linear regression. Do not assume the \(y\)-intercept is 0.

    Using the linear model, predict \(y\) when \(x=75\).


    Solution


  128. Question

    x y
    41.9 63.9
    46.0 67.3
    49.7 78.0
    42.9 61.5
    46.4 70.4
    43.5 62.6
    43.5 68.8
    48.2 78.6
    50.1 83.2

    Find the trendline using a simple linear regression. Do not assume the \(y\)-intercept is 0.

    Using the linear model, predict \(y\) when \(x=32\).


    Solution


  129. Question

    x y
    50.7 422.9
    60.3 505.4
    48.9 403.0
    57.4 457.6
    57.4 484.1
    60.9 520.3
    53.3 448.9
    48.9 406.4
    59.9 498.8

    Find the trendline using a simple linear regression. Do not assume the \(y\)-intercept is 0.

    Using the linear model, predict \(y\) when \(x=48\).


    Solution


  130. Question

    x y
    87.1 279.0
    89.0 284.0
    87.4 279.0
    85.7 277.9
    88.7 288.7
    88.6 288.9

    Find the trendline using a simple linear regression. Do not assume the \(y\)-intercept is 0.

    Using the linear model, predict \(y\) when \(x=97\).


    Solution